Determinant and Inverse Matrix - NYU Courant
[Pages:5]Determinant and Inverse Matrix
Liming Pang
Definition 1. A n ? n square matrix A is invertible if there exists a n ? n matrix A-1 such that AA-1 = A-1A = In, where In is the identity n ? n matrix. If A-1 exists, we say A-1 is the inverse matrix of A.
Proposition 2. If A and B are n ? n matrices, then AB = In BA = In.
Example 3.
83 52
2 -5
-3 8
=
1 0
0 1
=
2 -5
-3 8
83 52
So
8 5
3 2
is invertible and its inverse is
2 -5
-3 8
Remark 4. If A is invertible, then it follows directly from definition that A-1 is also invertible and the inverse of A-1 is A.
Proposition 5. If A, B are n ? n matrices, then:
1. (A-1)-1 = A
2. (AB)-1 = B-1A-1
3. (AT )-1 = (A-1)T
It is a natural question to ask if there is some way to tell if a matrix is invertible before looking for its inverse. It turns out that the concept of determinant solve the problem completely.
We will define determinant of a n ? n matrix in a recursive manner.
Definition 6. A is a n ? n square matrix, where n > 1. Define the matrix Aij to be the (n - 1) ? (n - 1) square matrix obtain from A by deleting the i-th row and j-th column.
1
2 -3 1
Example 7. Let A = -5 8 9 ,
4 7 -2
-5 9
2 -3
then A12 = 4 -2 , A33 = -5 8
Definition 8. If A = {a} is a 1 ? 1 matrix, define the determinant of A to be the number
det(A) = a
If A = {aij} is a n ? n (n > 1) square matrix, then define the determinant of A to be the number
n
det(A) = (-1)i+1ai1 det Ai1
i=1
We now take the case n = 2: If A = {aij} is a 2 ? 2 matrix, then bu definition,
2
det(A) = (-1)i+1ai1 det(Ai1) = a11 det A11 - a21 det A21 = a11a22 - a21a12
i=1
So we have obtained the formula
det
a c
b d
= ad - bc
(0.1)
We can thus directly apply Formula 0.1 when computing the determinant of a 2 ? 2 matrix.
Example 9.
det
1 4
-5 2
= 1 ? 2 - 4 ? (-5) = 22
Next let's compute the determinant of a 3 ? 3 matrix:
Example 10.
1 det 4
7
2 5 8
3 6 = 1 ? det 9
5 8
6 9
- 4 ? det
2 8
3 9
+ 7 ? det
2 5
3 6
= 1 ? (5 ? 9 - 6 ? 8) - 4 ? (2 ? 9 - 3 ? 8) + 7 ? (2 ? 6 - 3 ? 5) = 1 ? (-3) - 4 ? (-6) + 7 ? (-3) =0
2
There are some interesting properties regarding determinants: Proposition 11. Let A be an n ? n matrix, then:
1. det(In) = 1 2. det(A) = det(AT ).
3. If all entries in some row (column) are 0, then det(A) = 0.
4. If B is obtained from A by switching two rows (or columns), then det(B) = - det(A).
5. If B is obtained from A by multiplying some number to all the entries in one of the rows (columns), then det(B) = det(A).
6. If B is obtained from A by adding a multiple of one row (column) to another, then det(A) = det(B).
7. If there are two rows (columns) of A such that one is a multiple of another, then det(A) = 0
8. det is multiplicative: for any n ? n matrices A and B, det(AB) = det(A) det(B)
1 1 3
Example 12. By the above propositions, we see that det 4 4 6 = 0.
779
1 4 3
Then we add 1 times the third column to the second column, det 4 10 6 =
7 16 9
1 2 3
0.
Next
we
multiply
the
second
column
by
1 2
,
we
get
det 4
5
6
=
1 2
?0
=
789
0, which agrees with the previous example.
Remark 13. The above propositions implies we can compute determinant in a more general way:
n
n
det(A) = (-1)i+jaij det(Aij) = (-1)i+j det(Aij)
i=1
j=1
3
An important application of determinant is that it can be used to test if a square matrix is invertible or not:
Theorem 14. A square matrix A is invertible if and only if det(A) = 0.
1 2 3
1 2 3
Example 15. We have shown that det 4 5 6 = 0, so 4 5 6 is not
789
789
invertible.
Example 16. A 2 ? 2 matrix
a c
b d
is invertible if and only if
det(
a c
b d
) = ad - bc = 0
The method of Gaussian Elimination can be applied to computing the inverse of a given matrix.
Theorem 17. A is an invertible n ? n matrix. If we form the matrix
A In
a11 ... a1n 1 0 ... 0
=
a21 ... a2n 0 1 ... 0 .........................
an1 ... ann 0 0 ... 1
and apply the method of Gaussian Elimination, we will obtain
In A-1
Example 18. Find the inverse of
1 3 3 1 3 4
143
1 3 3 1 0 0 1 3 3 1 0 0 1 3 3 1 0 0
1 3 4 0 1 0 0 0 1 -1 1 0 0 1 0 -1 0 1
143001
0 1 0 -1 0 1
0 0 1 -1 1 0
1 0 3 4 0 -3 1 0 0 7 -3 -3
0 1 0 -1 0 1 0 1 0 -1 0 1
0 0 1 -1 1 0
0 0 1 -1 1 0
4
So we find the inverse of the given matrix is
7 -3 -3 -1 0 1
-1 1 0
When an invertible matrix is 2 ? 2, there is a formula for its inverse:
Proposition 19. If
a c
b d
is an invertible matrix, then its inverse is
1 d -b ad - bc -c a
Example 20. Find the inverse of
1 3
2 4
1 2 -1
1
4 -2 -2 1
34
= 1?4-2?3
-3
1
=
3 2
-
1 2
Proposition 21. For a n ? n diagonal matrix A = (aij) (i.e. aij = 0 for any i = j), its determinant is
Example 22.
det(A) = a11a22...ann
2 0 0 det 0 3 0 = 2 ? 3 ? (-4) = -24
0 0 -4
5
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- matrix inverses and determinants date period
- 17 vandermonde determinants
- the determinant a means to calculate volume
- lecture 18 properties of determinants
- chapter 8 matrices and determinants
- 5 3 determinants and cramer s rule university of utah
- determinant and inverse matrix nyu courant
- section 2 3 properties of determinants
- determinants and eigenvalues
- determinants inverse matrices
Related searches
- etf and inverse etf list
- roles and responsibilities matrix template
- determinant of a matrix calculator
- determinant of coefficient matrix calculator
- inverse matrix calculator
- inverse matrix calc
- find the determinant of the matrix calculator
- determinant of 3x3 matrix calc
- determinant of a matrix calc
- determinant of a matrix meaning
- find determinant of 2x2 matrix calculator
- system of equations inverse matrix calculator